# q-expansion of newform 198.4.a.c, downloaded from the LMFDB on 05 October 2026.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_data))]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 198
weight = 4
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[-2], [0], [8], [-22], [11], [-54], [26], [-38], [64], [-294], [36], [-390], [-138], [-242], [-132], [-388], [732], [430], [520], [-420], [-594], [506], [-380], [256], [418], [650], [-1248], [652], [1914], [1292], [-54], [2188], [284], [-2854], [-3202], [-302], [-394], [-1508], [2352], [362], [4064], [-2318], [4260], [-2822], [2506], [-5332], [146], [4800], [3300], [4310], [-5978], [4592], [6046], [-7168], [5144], [-5928], [1340], [254], [-2590], [-6658], [-1226], [-8858], [-5942], [7432], [2654], [-10512], [3812], [-162], [-8260], [2046], [-4584], [-5904], [-7768], [2062], [5732], [3128], [-336], [-14294], [2692], [-1154], [-9048], [-6670], [10368], [5726], [-12530], [-10172], [9416], [8614], [-11218], [-2328], [2436], [-15912], [-7828], [8132], [-9088], [8456], [11492], [-6132], [-706], [20434], [23702], [-13074], [-4796], [6538], [23326], [-1822], [13380], [19710], [-14256], [8794], [-12182], [16182], [19124], [-28180], [13780], [12144], [15644], [-13864], [6372], [-17028], [-4242], [-16894], [26290], [-15048], [10308], [-20514], [-34314], [-19020], [-32148], [-33310], [27638], [21160], [6092], [1586], [14226], [1546], [-38892], [-17810], [-26856], [1906], [3530], [15970], [-6520], [38756], [5686], [-29640], [40182], [15794], [-1316], [6968], [-12858], [22300], [22288], [50088], [-27164], [13740], [-9202], [-9732], [-21086], [-24870], [13104], [5898], [29314], [-20148], [46820], [-51576], [-14668], [-44978]]